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Date: Wed, 10 May 1995 22:38:32 -0400 (EDT)
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Re: more on the slice group
In-Reply-To: Message of 05/09/95 at 12:11:02 from hoey@AIC.NRL.Navy.Mil
On 05/09/95 at 12:11:02 hoey@AIC.NRL.Navy.Mil said:
>No, the 4-spot pattern is also a local maximum at 12 qtw, although its
>symmetry group is of order 16. Jim Saxe and I reported this on 22
>March 1981, in "No short relations and a new local maximum".
Argh! After Dan and Mike pointed this out, I did remember having seen
it in the archives. Worse still, Dan pointed it out again on
3 August 1992. But since it has come up, let's take a brief look
at the 22 March 1981 note.
> With five-qtw searches, it became possible to check another
>conjecture, using an approach that Jim suggested. The four-spot
>pattern
>
> U U U
> U U U
> U U U
>
> R R R B B B L L L F F F
> R L R B F B L R L F B F
> R R R B B B L L L F F F
>
> D D D
> D D D
> D D D
>
>is solvable in twelve qtw, either by (FFBB)(UD')(LLRR)(UD') or by its
>inverse, (DU')(LLRR)(DU')(FFBB). It is immediate that a twelve qtw
>path from this pattern to START can begin with a twist of any face in
>either direction. The program was used to verify that there are no ten
>qtw paths. (It generated the set of positions attainable at most five
>qtw from START and the set of positions obtainable from the four-spot
>in at most five qtw, and verified that the intersection of the two sets
>is empty.) Thus the four-spot is exactly twelve qtw from START and all
>its neighbors are exactly eleven qtw from START, proving that the
>four-spot is a local maximum.
>
Call the 4-spot s. Then, the twelve neighbors form two M-conjugacy
classes: N1={sL,sL',sF,sF',sR,sR',sB,sB'} and N2={sU,sU',sD,sD'}.
Also, we have s'=s. Dan and Jim's solution starts in N1 and ends with
a quarter-turn from N2, and since s'=s, we can say "or vice versa".
Hence, we can start a solution with any of the twelve quarter turns,
and therefore s is a local maximum.
There are other positions with the same symmetry characteristics as
the 4-spot. That is, there are other positions for which the
symmetry group contains sixteen elements. There are only three subgroups
of M containing sixteen elements, and the three subgroups are M conjugate.
The three M-conjugates of the 4-spot position correspond to the three
conjugate subgroups of M containing sixteen elements. But what of
other positions with the same symmetry group? For example, if the
edges of the 4-spot are all flipped, is the position a local maximum?
I don't know, but it is interesting to see how far we can get without
knowing a process.
Call the 4-spot with all edges flipped t. Then, we certainly have
t'=t. Is this true of all positions whose symmetry group contains
sixteen elements? Also, we certainly have the twelve neighbors
forming M-conjugacy classes similar to those for s, N1 with eight
elements and N2 with four. Is this true of all positions whose symmetry
group contains sixteen elements? Finally, a solution either starts in
N1 or starts in N2. If starting in N1 implies ending with a
quarter-turn from N2 or vice versa, then t is a local maximum.
Can we prove such a thing without actually finding a solution?
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) (304) 293-5192
Associate Director, WVNET (304) 293-5540 fax
837 Chestnut Ridge Road BRYAN@WVNVM
Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU